Elementary Number Theory Problems 3.3 Solution (David M. Burton's 7th Edition) - Q12

Background

All theorems, corollaries, and definitions listed in the book's order:

I will only use theorems or facts that are proved before this question. So you will not see that I quote theorems or facts from the later chapters.

Question

Let $p_{n}$ denote the $n$th prime number. For $n \geq 3$, prove that $p_{n+3}^{2} < p_{n}p_{n+1}p_{n+2}$.
[Hint: Note that $p_{n+3}^{2} < 4p_{n+2}^2 <8p_{n+1}p_{n+2}$.]

Solution

For this question, we need to use the theorem from Bertrand's Postulate mentioned in Chapter 3.2, $p_{n+1} < 2p_{n}$.